WASHINGTON—Lawmakers on Wednesday voiced concern over the Obama administration's plans for monitoring individuals who hold security clearances as part of a crackdown in the wake of leaks by former government contractor Edward Snowden.

Sens. Charles Grassley (R., Iowa) and Ron Wyden (D., Ore.) said in a letter to Director of National Intelligence James Clapper that the stated possibility of "continuous evaluation" of legislative officials would raise constitutional questions and that extensive employee monitoring...